How to cut at vertex position of another object?
If the plane and cube belong to the same mesh, in Edit mode, you can use CtrlF Face Menu &gt; Intersect (Knife). It has 3 modes:

(Illustrations exploded to show separate parts)

All: all the results of intersection are split to separate islands
Cut: Edges are created, but not split, at intersections in all parts
Merge: Edges are created at intersections, all vertices merged, resulting here in an internal face.